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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh8aa | Structured version Visualization version GIF version |
Description: Part of Part (8) in [Baer] p. 48. (Contributed by NM, 12-May-2015.) |
Ref | Expression |
---|---|
mapdh8a.h | β’ π» = (LHypβπΎ) |
mapdh8a.u | β’ π = ((DVecHβπΎ)βπ) |
mapdh8a.v | β’ π = (Baseβπ) |
mapdh8a.s | β’ β = (-gβπ) |
mapdh8a.o | β’ 0 = (0gβπ) |
mapdh8a.n | β’ π = (LSpanβπ) |
mapdh8a.c | β’ πΆ = ((LCDualβπΎ)βπ) |
mapdh8a.d | β’ π· = (BaseβπΆ) |
mapdh8a.r | β’ π = (-gβπΆ) |
mapdh8a.q | β’ π = (0gβπΆ) |
mapdh8a.j | β’ π½ = (LSpanβπΆ) |
mapdh8a.m | β’ π = ((mapdβπΎ)βπ) |
mapdh8a.i | β’ πΌ = (π₯ β V β¦ if((2nd βπ₯) = 0 , π, (β©β β π· ((πβ(πβ{(2nd βπ₯)})) = (π½β{β}) β§ (πβ(πβ{((1st β(1st βπ₯)) β (2nd βπ₯))})) = (π½β{((2nd β(1st βπ₯))π β)}))))) |
mapdh8a.k | β’ (π β (πΎ β HL β§ π β π»)) |
mapdh8aa.f | β’ (π β πΉ β π·) |
mapdh8aa.mn | β’ (π β (πβ(πβ{π})) = (π½β{πΉ})) |
mapdh8aa.eg | β’ (π β (πΌββ¨π, πΉ, πβ©) = πΊ) |
mapdh8aa.ee | β’ (π β (πΌββ¨π, πΉ, πβ©) = πΈ) |
mapdh8aa.x | β’ (π β π β (π β { 0 })) |
mapdh8aa.y | β’ (π β π β (π β { 0 })) |
mapdh8aa.z | β’ (π β π β (π β { 0 })) |
mapdh8aa.zt | β’ (π β (πβ{π}) β (πβ{π})) |
mapdh8aa.t | β’ (π β π β (π β { 0 })) |
mapdh8aa.yn | β’ (π β Β¬ π β (πβ{π, π})) |
mapdh8aa.xn | β’ (π β Β¬ π β (πβ{π, π})) |
Ref | Expression |
---|---|
mapdh8aa | β’ (π β (πΌββ¨π, πΊ, πβ©) = (πΌββ¨π, πΈ, πβ©)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh8a.h | . . 3 β’ π» = (LHypβπΎ) | |
2 | mapdh8a.u | . . 3 β’ π = ((DVecHβπΎ)βπ) | |
3 | mapdh8a.v | . . 3 β’ π = (Baseβπ) | |
4 | mapdh8a.s | . . 3 β’ β = (-gβπ) | |
5 | mapdh8a.o | . . 3 β’ 0 = (0gβπ) | |
6 | mapdh8a.n | . . 3 β’ π = (LSpanβπ) | |
7 | mapdh8a.c | . . 3 β’ πΆ = ((LCDualβπΎ)βπ) | |
8 | mapdh8a.d | . . 3 β’ π· = (BaseβπΆ) | |
9 | mapdh8a.r | . . 3 β’ π = (-gβπΆ) | |
10 | mapdh8a.q | . . 3 β’ π = (0gβπΆ) | |
11 | mapdh8a.j | . . 3 β’ π½ = (LSpanβπΆ) | |
12 | mapdh8a.m | . . 3 β’ π = ((mapdβπΎ)βπ) | |
13 | mapdh8a.i | . . 3 β’ πΌ = (π₯ β V β¦ if((2nd βπ₯) = 0 , π, (β©β β π· ((πβ(πβ{(2nd βπ₯)})) = (π½β{β}) β§ (πβ(πβ{((1st β(1st βπ₯)) β (2nd βπ₯))})) = (π½β{((2nd β(1st βπ₯))π β)}))))) | |
14 | mapdh8a.k | . . 3 β’ (π β (πΎ β HL β§ π β π»)) | |
15 | mapdh8aa.eg | . . . 4 β’ (π β (πΌββ¨π, πΉ, πβ©) = πΊ) | |
16 | mapdh8aa.f | . . . . 5 β’ (π β πΉ β π·) | |
17 | mapdh8aa.mn | . . . . 5 β’ (π β (πβ(πβ{π})) = (π½β{πΉ})) | |
18 | mapdh8aa.x | . . . . 5 β’ (π β π β (π β { 0 })) | |
19 | mapdh8aa.y | . . . . . 6 β’ (π β π β (π β { 0 })) | |
20 | 19 | eldifad 3953 | . . . . 5 β’ (π β π β π) |
21 | 1, 2, 14 | dvhlvec 40483 | . . . . . . 7 β’ (π β π β LVec) |
22 | 18 | eldifad 3953 | . . . . . . 7 β’ (π β π β π) |
23 | mapdh8aa.z | . . . . . . . 8 β’ (π β π β (π β { 0 })) | |
24 | 23 | eldifad 3953 | . . . . . . 7 β’ (π β π β π) |
25 | mapdh8aa.xn | . . . . . . 7 β’ (π β Β¬ π β (πβ{π, π})) | |
26 | 3, 6, 21, 22, 20, 24, 25 | lspindpi 20979 | . . . . . 6 β’ (π β ((πβ{π}) β (πβ{π}) β§ (πβ{π}) β (πβ{π}))) |
27 | 26 | simpld 494 | . . . . 5 β’ (π β (πβ{π}) β (πβ{π})) |
28 | 10, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 16, 17, 18, 20, 27 | mapdhcl 41101 | . . . 4 β’ (π β (πΌββ¨π, πΉ, πβ©) β π·) |
29 | 15, 28 | eqeltrrd 2826 | . . 3 β’ (π β πΊ β π·) |
30 | 10, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 16, 17, 18, 19, 29, 27 | mapdheq 41102 | . . . . 5 β’ (π β ((πΌββ¨π, πΉ, πβ©) = πΊ β ((πβ(πβ{π})) = (π½β{πΊ}) β§ (πβ(πβ{(π β π)})) = (π½β{(πΉπ πΊ)})))) |
31 | 15, 30 | mpbid 231 | . . . 4 β’ (π β ((πβ(πβ{π})) = (π½β{πΊ}) β§ (πβ(πβ{(π β π)})) = (π½β{(πΉπ πΊ)}))) |
32 | 31 | simpld 494 | . . 3 β’ (π β (πβ(πβ{π})) = (π½β{πΊ})) |
33 | mapdh8aa.ee | . . . 4 β’ (π β (πΌββ¨π, πΉ, πβ©) = πΈ) | |
34 | mapdh8aa.t | . . . . . . 7 β’ (π β π β (π β { 0 })) | |
35 | 34 | eldifad 3953 | . . . . . 6 β’ (π β π β π) |
36 | mapdh8aa.yn | . . . . . 6 β’ (π β Β¬ π β (πβ{π, π})) | |
37 | 3, 6, 21, 20, 24, 35, 36 | lspindpi 20979 | . . . . 5 β’ (π β ((πβ{π}) β (πβ{π}) β§ (πβ{π}) β (πβ{π}))) |
38 | 37 | simpld 494 | . . . 4 β’ (π β (πβ{π}) β (πβ{π})) |
39 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 15, 33, 38, 25, 18, 19, 23 | mapdh75d 41128 | . . 3 β’ (π β (πΌββ¨π, πΊ, πβ©) = πΈ) |
40 | mapdh8aa.zt | . . 3 β’ (π β (πβ{π}) β (πβ{π})) | |
41 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 29, 32, 39, 19, 23, 40, 34, 36 | mapdh8a 41149 | . 2 β’ (π β (πΌββ¨π, πΈ, πβ©) = (πΌββ¨π, πΊ, πβ©)) |
42 | 41 | eqcomd 2730 | 1 β’ (π β (πΌββ¨π, πΊ, πβ©) = (πΌββ¨π, πΈ, πβ©)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2932 Vcvv 3466 β cdif 3938 ifcif 4521 {csn 4621 {cpr 4623 β¨cotp 4629 β¦ cmpt 5222 βcfv 6534 β©crio 7357 (class class class)co 7402 1st c1st 7967 2nd c2nd 7968 Basecbs 17149 0gc0g 17390 -gcsg 18861 LSpanclspn 20814 HLchlt 38723 LHypclh 39358 DVecHcdvh 40452 LCDualclcd 40960 mapdcmpd 40998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-riotaBAD 38326 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-ot 4630 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13486 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-0g 17392 df-mre 17535 df-mrc 17536 df-acs 17538 df-proset 18256 df-poset 18274 df-plt 18291 df-lub 18307 df-glb 18308 df-join 18309 df-meet 18310 df-p0 18386 df-p1 18387 df-lat 18393 df-clat 18460 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-grp 18862 df-minusg 18863 df-sbg 18864 df-subg 19046 df-cntz 19229 df-oppg 19258 df-lsm 19552 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-drng 20585 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lvec 20947 df-lsatoms 38349 df-lshyp 38350 df-lcv 38392 df-lfl 38431 df-lkr 38459 df-ldual 38497 df-oposet 38549 df-ol 38551 df-oml 38552 df-covers 38639 df-ats 38640 df-atl 38671 df-cvlat 38695 df-hlat 38724 df-llines 38872 df-lplanes 38873 df-lvols 38874 df-lines 38875 df-psubsp 38877 df-pmap 38878 df-padd 39170 df-lhyp 39362 df-laut 39363 df-ldil 39478 df-ltrn 39479 df-trl 39533 df-tgrp 40117 df-tendo 40129 df-edring 40131 df-dveca 40377 df-disoa 40403 df-dvech 40453 df-dib 40513 df-dic 40547 df-dih 40603 df-doch 40722 df-djh 40769 df-lcdual 40961 df-mapd 40999 |
This theorem is referenced by: mapdh8ab 41151 |
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